There are many examples throughout mathematics of abstracting the formal properties of a "familiar" structure, but then having a theorem stating that all models of the abstract axioms embed into one of the original "familiar" structures. Examples include
- every group is a subgroup of a symmetric group;
- every small category is concretizable (essentially by the Yoneda lemma);
- every manifold smoothly embeds into real space;
- every abelian category is a full subcategory of a category of modules, with exact inclusion;
- every topos can be embedded in a Boolean one;
- ...
Is there such an embedding theorem for [some nice subcategory of] the category of topological spaces? Here I am not sure what the "familiar" structures would be.
More generally, can we formulate a metatheorem on when "every object behaving formally like a concrete structure embeds into one"?